Dynamic Systems and Feedback

Introduction Structure of a closed-loop control system Example: Temperature Control in Shower Problems Block Diagrams Common blocks, continuous-time Example Summary Problems Mathematical Modeling and Simulation Systems of 1st order, coupled differential equations Remarks about Integration Options in simulink Problems State Variables Definition of State State-variables: from first order evolution equations State-variables: from a block diagram Problems First Order, Linear, Time-Invariant (LTI) ODE The Big Picture Solution of a First Order LTI ODE Free response Forced response, constant inputs Forced response, bounded inputs Stable system, Forced response, input approaching 0 Linearity Forced response, input approaching a constant limit Forced response, Sinusoidal inputs Forced response, input approaching a Sinusoid First-order delay-differential equation: Stability Summary Problems Feedback systems First-order plant, Proportional control Proportional Plant, first-order controller Problems Two forms of high-order Linear ODEs, with forcing Problems Jacobian Linearizations, equilibrium points Jacobians and the Taylor Theorem Equilibrium Points Deviation Variables Tank Example Output Equations Calculus for systems not in standard form Another common non-standard form Linearizing about general solution Problems Additional Related Problems Linear Algebra Review Notation Determinants Inverses Solving Linear equations: Gaussian Elimination Matrix functions of Time Linear Systems and Time-Invariance Linearity of solution Time-Invariance Matrix Exponential Diagonal A Block Diagonal A Effect of Similarity Transformations Solution To State Equations Examples Problems Eigenvalues, eigenvectors, stability Diagonalization: Motivation Eigenvalues Diagonalization Procedure eAt as t Complex Eigenvalues Alternate parametrization with complex eigenvalues Examples Step response Quick estimate of unit-step-response of 2nd order system Problems Frequency Response for Linear Systems: State-Space representations Theory for Stable System: Complex Input Signal MIMO Systems: Response due to real sinusoidal inputs Experimental Determination Steady-State response Important special cases for designing closed-loop systems Roots of 2nd-order monic polynomial Setting the coefficients to attain certain roots 1st-order plant, 1st-order controller 2nd-order plant, constant-gain controller with derivative feedback Problems Step response Quick estimate of unit-step-response of 2nd order system Stabilization by State-Feedback Theory State-Feedback with Integral Control Theory Example Problems Linear-quadratic Optimal Control Learning more Single, high-order, linear ODES (SLODE) Linear, Time-Invariant Differential Equations Importance of Linearity Solving Homogeneous Equation Interpretation of complex roots to ODEs with real-coefficients General Solution Technique Behavior of Homogeneous Solutions as t Response of stable system to constant input (Steady-State Gain) Example Stability Conditions for 2nd order differential equation Important 2nd order example Summary for SLODEs Main Results General Solution Technique Behavior of Homogeneous Solutions as t Stability of Polynomials 2nd order differential equation Solutions of 2nd order differential equation Problems Frequency Responses of Linear Systems Complex and Real Particular Solutions Response due to real sinusoidal inputs Problems Derivatives appearing on the inputs: Effect on the forced response Introduction Other Particular Solutions Limits approaching steps Problems Distributions Introduction Procedure to get step response Summary: Solution of SLODEs with Derivatives on the inputs Introduction Problems Transfer functions Linear Differential Operators (LDOs) Algebra of Linear differential operations Feedback Connection More General Feedback Connection Cascade (or Series) Connection Parallel Connection General Connection Systems with multiple inputs Poles and Zeros of Transfer Functions Problems Arithmetic of Feedback Loops Tradeoffs Signal-to-Noise ratio What's missing? Problems Robustness Margins Gain Margin Time-Delay Margin Percentage Variation Margin The Small-Gain Theorem Necessary and Sufficient Version Application to Percentage Variation Margin Summary Examples Generic Missile Application to percentage variation margin Problems Gain/Time-delay margins: Alternative derivation Sylvester's determinant identity Setup Gain margin Time delay margin Appendix Connection between Frequency Responses and Transfer functions Interconnections Decomposing Systems into Simple Parts Problems Unfiled problems Recent exams Fall 2017 Midterm 1 Fall 2017 Midterm 2 Fall 2017 Final Fall 2015 Midterm 1 Fall 2015 Midterm 2 Fall 2015 Final Spring 2014, Midterm 1 Spring 2014, Midterm 2 Spring 2014 Final Exam Older exams Spring 2012, Midterm 1 Spring 2012, Midterm 2 Spring 2012, Final Exam Spring 2009, Midterm 1 Spring 2009, Midterm 2 Spring 2009, Final Exam Spring 2005 Midterm 1 Spring 2005 Midterm 2 Spring 2004 Midterm 1 Fall 2003 Midterm 1 Fall 2003 Midterm 2 Fall 2003 Final